A quantum graph is a metric graph equipped with Schrödinger operators acting on edges and some conditions at vertices (see, for instance, [1]}). For quantum graphs, there is a wide variety of conditions coupling the wave functions at vertices, in contrast to quantum systems with point interactions on the line.
This makes the theory of Schrödinger operators on graphs much richer.
However, the rich set of possible vertex coupling conditions complicates the construction of solvable models for specific quantum processes because the problem of choosing physically motivated conditions arises.
The mathematical approach to the construction of such models, in addition to the experimental one, consists in different approximations of Hamiltonians on graphs. The approximation of a graph by thin tubular domains
is the most natural. The problem then is to analyse the convergence of 3D Hamiltonians as the tubular networks shrink to the graph and obtain the limit point interactions at the vertices [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
Another way of finding physically motivated point interactions is the approximation of quasi-one-dimensional meso- or nano-scale systems by Schrödinger operators on a graph with potentials singularly perturbed in a neighbourhood of the vertices [9]}, [10]}, [11]}, [12]}, [13]}. Then the limit Hamiltonian, obtained in a good operator topology, is the desired solvable model of the system.
Also interesting is the inverse problem, which consists in constructing approximations of given point interactions at a vertex by some regular Hamiltonians [14]}, [15]}, [16]}. It has been proven in [16]} that any singular vertex coupling can be approximated by a family of quantum graphs with simultaneous perturbation of the geometry of the graph and the potentials.
